Slides09-2010 - Moments of More that One Random Variable:...

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Moments of More that One Random Variable: Lecture IX Charles B. Moss September 10, 2010 Charles B. Moss () Moments of Multiple Random Variables September 10, 2010 1 / 16
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1 Covariance and Correlation 2 Conditional Mean and Variance Charles B. Moss () Moments of Multiple Random Variables September 10, 2010 2 / 16
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Covariance and Correlation Defnition 4.3.1 The covariance between two random variables X and Y can be defned as Cov ( X , Y )= E [( X E [ X ]) ( Y E [ Y ])] = E [ XY X E [ Y ] E [ x ] Y + E [ X ][ Y ]] = E [ XY ] E [ X ] E [ Y ] E [ X ] E [ Y ]+ E [ X ] E [ Y ] = E [ XY ] E [ X ] E [ Y ] (1) I Note that this is simply a generalization oF the standard variance Formulation. Specifcally, letting Y X yields Cov ( XX E [ XX ] E [ X ] E [ X ] = E ± X 2 ² ( E [ X ]) 2 (2) I ±rom a sample perspective, we have V ( X 1 N N i =1 x 2 i ¯ x 2 Cov ( X , Y 1 N N i =1 x i y i ¯ x ¯ y (3) Charles B. Moss () Moments oF Multiple Random Variables September 10, 2010 3 / 16
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Continued I Together the variance and covariance matrices are typically written as a variance matrix Σ= ± V ( X ) Cov ( X , Y ) Cov ( Y , X ) V ( Y ) ² = ± σ xx σ xy σ yx σ yy ² (4) Note that Cov ( X , Y )= σ xy = σ yx = Cov ( Y , X ). I Substituting the sample measures into the variance matrix yields S = ± s xx s xy s yx s yy ² = " 1 N N i =1 x i x i ¯ x ¯ x 1 N N i =1 x i y i ¯ x ¯ y 1 N N i =1 y i x i ¯ y ¯ x 1 N N i =1 y i y i ¯ y ¯ y # = 1 N " i =1 x i x i N i =1 x i y i N i =1 y i x i N i =1 y i y i # ± ¯ x ¯ x ¯ x ¯ y ¯ y ¯ x ¯ y ¯ y ² (5) The sample covariance matrix can then be written as S = 1 N ± x 1 ··· x N y 1 y N ² ± ¯ x ¯ y ² ³ ¯ x ¯ y ´ . (6) Charles B. Moss () Moments of Multiple Random Variables September 10, 2010 4 / 16
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Continued I In terms of the theoretical distribution, the variance matrix can be written as Σ= " R −∞ R −∞ ( x μ x
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Slides09-2010 - Moments of More that One Random Variable:...

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